The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so called Navier-Lam\'e system is considered. Such a system introduces the displacement, rotation and pressure of some linear and elastic structure. The analysis of the spectral problem is based in the compact operators theory. A finite element method based in polynomials of degree $k\geq 1$ are considered in order to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimate are presented. An a posteriori error analysis is performed, where the reliability and efficiency of the proposed estimator is proved. We end this contribution reporting a series of numerical tests in order to assess the performance of the proposed numerical method, for the a priori and a posteriori estimates.

翻译：本文件介绍在考虑所谓的纳维埃-Lam\'e系统时对弹性方程的精度问题的分析。这种系统将引入某些线性和弹性结构的移位、旋转和压力。光谱问题的分析以紧凑操作员理论为基础。以1美元(美元)1美元(美元)的多元数值为基础的有限元素法是考虑的,以便接近系统的精度和元件。提出了一致和误差估计。进行了事后误差分析,证明了拟议估算器的可靠性和效率。我们结束这一分析,报告一系列数字测试,以便评估拟议数字方法的性能,用于先验和后验估计。